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The aim of this paper is two-fold: Firstly, we prove Toda's$\chi$-independence conjecture for Gopakumar--Vafa invariants of arbitrarylocal curves. Secondly, following Davison's work, we introduce the BPScohomology for moduli spaces of Higgs bundles of rank $r$ and Eulercharacteristic $\chi$ which are not necessary coprime, and show that it doesnot depend on $\chi$. This result extends the Hausel--Thaddeus conjecture onthe $\chi$-independence of E-polynomials proved by Mellit,Groechenig--Wyss--Ziegler and Yu in two ways: we obtain an isomorphism of mixedHodge modules on the Hitchin base rather than an equality of E-polynomials, andwe do not need the coprime assumption.  The proof of these results is based on a description of the moduli stack ofone-dimensional coherent sheaves on a local curve as a global critical locuswhich is obtained in the companion paper by the first author and Naruki Masuda.

Cohomological $\chi$-independence for Higgs bundles and Gopakumar-Vafa invariants